Question

Solving a First-Order Linear Differential Equation In Exercises \(5-14,\) solve the first-order linear differential equation. $$ y^{\prime}+3 y=e^{3 x} $$

Short answer

The final solution of the first order linear differential equation \(y^{\prime}+3 y=e^{3 x}\) is \(y(x) = \frac{1}{3}e^{3x}+Ce^{-3x}\).

Step-by-step solution

Identifying the integral

It's necessary to identify the integral factor for this equation. In a standard first order linear differential equation, the integrating factor can be found by the equation \(e^{\int p(x) dx}\). In this case, \(p(x)\) is 3, so the integrating factor is \(e^{\int 3 dx}\), which simplifies to \(e^{3x}\).

Multiplying the entire equation by the integral factor

Now, multiply every term of the differential equation by the integrating factor from step 1. The equation becomes \(e^{3x} y^{\prime}+3 e^{3x}y=e^{6x}\). The left-hand side of this equation is now the derivative of \(( e^{3x} y)\) with respect to \(x\).

Integrating both sides

To isolate \(y\), you need to perform an integration to both sides of the equation. So, integrate both sides with respect to \(x\): \(\int (e^{3x} y)' dx = \int e^{6x} dx\). This will give you \(e^{3x} y = \frac{1}{3}e^{6x}+ C\).

Solve for y

Now, we should solve for \(y\). Rearranging the equation obtained in step 3 for \(y\) by dividing both sides by \(e^{3x}\) gives us the function \(y(x)= \frac{1}{3}e^{3x}+Ce^{-3x}\).

Key concepts

Integrating Factor

An integrating factor is a function that is used to convert a non-exact differential equation into an exact one, making it easier to solve. In the context of first-order linear differential equations, which take the form
\(y^{\'prime} + p(x)y = g(x)\), an integrating factor is often the key to finding a solution. To find the integrating factor, we typically use the formula \(e^{\text{\textint} p(x) dx}\), where \(p(x)\) is the coefficient of \(y\) in the differential equation.

In the given exercise, with the differential equation \(y^{\'prime}+3y=e^{3x}\), the function \(p(x)\) equals 3. Calculating the integrating factor involves integrating \(p(x)\):
\(e^{\text{\textint} 3 dx} = e^{3x}\).
Once the integrating factor is determined, it is used to multiply every term of the differential equation. This operation turns the left-hand side into the derivative of a product, in this case, \(e^{3x} y\), which simplifies the equation significantly and brings us one step closer to solving for \(y\).

Why Use Integrating Factors?

Integrating factors simplify the structure of differential equations by transforming them into a form that allows for straightforward integration. This method streamlines the process of solving, especially for equations that might otherwise be too complex to handle with basic integration techniques.

When to Apply the Integrating Factor

An integrating factor is typically used when confronting a first-order linear differential equation where direct integration isn't possible due to the presence of the variable-dependent \(y\) term. By involving an integrating factor, we can neutralize the effects of \(p(x)y\) and allow for integration to proceed.

Differential Equation Solution

Finding the solution to a differential equation involves determining a function or set of functions that satisfy the equation. Specifically, for first-order linear differential equations, the solution process often includes obtaining an integrating factor, multiplying it across the equation, and then integrating both sides to isolate the dependent variable \(y\).

In our exercise, after multiplying the original equation by the integrating factor \(e^{3x}\), the left-hand side becomes \(e^{3x} y^{\'prime} + 3 e^{3x} y\), which is recognized as the derivative of \(e^{3x} y\) with respect to \(x\). This key insight significantly tightens the equation, allowing us to integrate both sides with ease and thus find the explicit formula for \(y\).

Finding the Constant of Integration

During integration, a constant of integration, denoted by \(C\), emerges. This constant represents an infinite number of solutions, as each value for \(C\) would satisfy the differential equation. The particular solution is often found by plugging in initial conditions if they are provided.

Expressing the Final Solution

Once integration is complete and the constant is accounted for, the last step is to express \(y\) explicitly in terms of \(x\). In the example given, the solution to the equation is \(y(x) = \frac{1}{3}e^{3x} + Ce^{-3x}\), where \(C\) is the constant of integration. It's crucial to remember that the solution to a differential equation is typically a family of functions, distinguished by different values of this constant.

Integration of Differential Equations

Integration is a fundamental technique used to solve differential equations, converting the equations into a more manageable form. For first-order linear differential equations, the process usually involves integrating both sides of the equation after multiplying by an integrating factor.

In the specific case of our exercise, integrating the modified equation \(\text{\textint} (e^{3x} y^{\'prime} + 3 e^{3x} y) dx = \text{\textint} e^{6x} dx\) simplifies the process as we are dealing with a recognized form of a derivative on the left. The integration of the left side effectively 'undoes' the derivative, leaving us with \(e^{3x} y\), and the right side integrates to \(\frac{1}{6}e^{6x} + C\), where \(C\) is the constant of integration.

Integral Calculus in Differential Equations

Integral calculus plays a crucial role in differential equations, often used to find antiderivatives or areas that are essential in identifying the solutions. In contexts like the exercise, it enables us to move from a derivative to the original function (antiderivative).

Techniques for Integration

There are various techniques for integration, such as substitution or integration by parts. However, in the context of solving linear differential equations, the integration is greatly facilitated by the correct use of an integrating factor, as it reduces the equation to a straightforward integral. Understanding when and how to apply these techniques is crucial for students tackling differential equations.